A200838 T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases

8, 25, 16, 56, 69, 32, 105, 194, 191, 64, 176, 435, 676, 529, 128, 273, 846, 1817, 2356, 1465, 256, 400, 1491, 4108, 7587, 8210, 4057, 512, 561, 2444, 8239, 19930, 31677, 28610, 11235, 1024, 760, 3789, 15128, 45465, 96690, 132263, 99700, 31113, 2048

Offset

1,1

Comments

Table starts

....8.....25......56......105.......176........273........400.........561

...16.....69.....194......435.......846.......1491.......2444........3789

...32....191.....676.....1817......4108.......8239......15128.......25953

...64....529....2356.....7587.....19930......45465......93472......177381

..128...1465....8210....31677.....96690.....250913.....577660.....1212729

..256...4057...28610...132263....469116....1384813....3570086.....8291391

..512..11235...99700...552247...2276028....7642875...22063924....56687801

.1024..31113..347434..2305835..11042700...42181611..136360286...387572529

.2048..86161.1210736..9627715..53576350..232803603..842739040..2649819955

.4096.238605.4219166.40199277.259938722.1284861277.5208328180.18116728573

Links

R. H. Hardin, Table of n, a(n) for n = 1..9999

Formula

Empirical for columns:

k=1: a(n) = 2*a(n-1)

k=2: a(n) = 3*a(n-1) -a(n-2) +a(n-3)

k=3: a(n) = 4*a(n-1) -2*a(n-2) +a(n-3) -a(n-4)

k=4: a(n) = 5*a(n-1) -4*a(n-2) +3*a(n-3) -3*a(n-4) +a(n-5) -a(n-6)

k=5: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7)

k=6: a(n) = 7*a(n-1) -9*a(n-2) +6*a(n-3) -9*a(n-4) +7*a(n-5) -7*a(n-6) +5*a(n-7) -2*a(n-8) +a(n-9)

k=7: a(n) = 8*a(n-1) -12*a(n-2) +6*a(n-3) -10*a(n-4) +12*a(n-5) -11*a(n-6) +11*a(n-7) -6*a(n-8) +3*a(n-9) -a(n-10)

Empirical for rows:

n=1: a(k) = (2/3)*k^3 + 3*k^2 + (10/3)*k + 1

n=2: a(k) = (5/12)*k^4 + (19/6)*k^3 + (79/12)*k^2 + (29/6)*k + 1

n=3: a(k) = (4/15)*k^5 + (17/6)*k^4 + (28/3)*k^3 + (73/6)*k^2 + (32/5)*k + 1

n=4: a(k) = (61/360)*k^6 + (93/40)*k^5 + (779/72)*k^4 + (521/24)*k^3 + (1801/90)*k^2 + (239/30)*k + 1

n=5: a(k) = (34/315)*k^7 + (163/90)*k^6 + (1981/180)*k^5 + (557/18)*k^4 + (7807/180)*k^3 + (1361/45)*k^2 + (333/35)*k + 1

n=6: a(k) = (277/4032)*k^8 + (1375/1008)*k^7 + (4933/480)*k^6 + (2723/72)*k^5 + (14161/192)*k^4 + (11197/144)*k^3 + (216211/5040)*k^2 + (929/84)*k + 1

n=7: a(k) = (124/2835)*k^9 + (1123/1120)*k^8 + (244/27)*k^7 + (1991/48)*k^6 + (57133/540)*k^5 + (74183/480)*k^4 + (291427/2268)*k^3 + (9739/168)*k^2 + (568/45)*k + 1

Example

Some solutions for n=4 k=3
..1....2....3....0....1....1....2....1....3....3....3....1....2....0....1....1
..0....0....0....2....1....0....3....3....1....3....0....3....2....3....1....0
..0....0....2....2....0....3....0....0....1....2....1....3....2....0....1....1
..3....0....1....3....3....3....3....2....1....2....0....1....2....0....0....1
..3....3....3....0....3....0....1....2....1....1....3....3....3....2....2....3
..1....3....2....0....1....3....3....2....2....1....0....1....2....1....1....0

Crossrefs

Column 1 is A000079(n+2)

Column 2 is A098182(n+3)

Row 1 is A131423(n+1)

Sequence in context: A023056 A103954 A217012 * A302160 A122984 A254341

Adjacent sequences: A200835 A200836 A200837 * A200839 A200840 A200841

Keyword

nonn,tabl

Author

R. H. Hardin Nov 23 2011

Status

approved